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<div  id='write'  class = 'is-mac first-line-indent'><h1><a name="考研数学基础汇总" class="md-header-anchor"></a><span>考研数学基础汇总</span></h1><blockquote><p><span>Author:张一极</span></p><p><span>个人博客:</span><a href='http://likedge.top'><span>极度空间</span></a></p><p><span>公众号:视觉迷航</span></p><p><span>本文适用于有基础,但不够全面作补充用,如有问题,欢迎斧正:zk@likedge.top</span></p><hr /><p><span>数列:</span></p><p><span>常见数列求和</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n38" cid="n38" mdtype="math_block">
			
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y="0"></use><use transform="scale(0.707)" xlink:href="#E43-MJMAIN-31" x="2367" y="0"></use><use transform="scale(0.707)" xlink:href="#E43-MJMAIN-29" x="2867" y="0"></use></g></g></g><use xlink:href="#E43-MJMAIN-3D" x="20332" y="0"></use><g transform="translate(21110,0)"><g transform="translate(397,0)"><rect stroke="none" width="1447" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E43-MJMATHI-6E" x="723" y="591"></use><g transform="translate(60,-382)"><use transform="scale(0.707)" xlink:href="#E43-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E43-MJMAIN-2B" x="600" y="0"></use><use transform="scale(0.707)" xlink:href="#E43-MJMAIN-31" x="1378" y="0"></use></g></g></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-23">\begin{array}{l}\sum_{k=1}^{n} k=1+2+3+\cdots+n=\frac{n(n+1)}{2} \\ \sum_{k=1}^{n} k^{2}=1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6} \\ \sum_{k=1}^{n} \frac{1}{k(k+1)}=\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}\end{array}</script>
					</div></div></blockquote><blockquote><p><span>三角函数:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n6" cid="n6" mdtype="math_block">
			
		<div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1">
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					</div></div></blockquote><p><span>三角变换:</span></p><blockquote><p><span>倍角公式和三角函数基本变换</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n21" cid="n21" mdtype="math_block">
			
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x="7522" y="0"></use><use xlink:href="#E75-MJMAIN-3C" x="8372" y="0"></use><use xlink:href="#E75-MJMATHI-3C0" x="9428" y="0"></use><use xlink:href="#E75-MJMAIN-29" x="10001" y="0"></use><use xlink:href="#E75-MJMAIN-2C" x="10390" y="0"></use><g transform="translate(10834,0)"><g transform="translate(250,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(53.596) matrix(1 0 0 -1 0 0)">则</text></g></g><g transform="translate(12164,0)"><use xlink:href="#E75-MJMAIN-73"></use><use xlink:href="#E75-MJMAIN-69" x="394" y="0"></use><use xlink:href="#E75-MJMAIN-6E" x="672" y="0"></use></g><use xlink:href="#E75-MJMATHI-78" x="13558" y="0"></use><use xlink:href="#E75-MJMAIN-3D" x="14408" y="0"></use><g transform="translate(15186,0)"><g transform="translate(397,0)"><rect stroke="none" width="2867" height="60" x="0" y="220"></rect><g transform="translate(897,676)"><use xlink:href="#E75-MJMAIN-32" x="0" y="0"></use><use xlink:href="#E75-MJMATHI-75" x="500" 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xlink:href="#E75-MJMATHI-75" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E75-MJMAIN-32" x="808" y="513"></use></g></g><g transform="translate(60,-734)"><use xlink:href="#E75-MJMAIN-31" x="0" y="0"></use><use xlink:href="#E75-MJMAIN-2B" x="722" y="0"></use><g transform="translate(1722,0)"><use xlink:href="#E75-MJMATHI-75" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E75-MJMAIN-32" x="808" y="408"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-41">u=\tan \frac{x}{2}(-\pi<x<\pi), \text { 则 } \sin x=\frac{2 u}{1+u^{2}}, \cos x=\frac{1-u^{2}}{1+u^{2}}</script>
					</div></div></blockquote><blockquote><p><span>因式分解</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n65" cid="n65" mdtype="math_block">
			
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transform="scale(0.707)" xlink:href="#E122-MJMAIN-33" x="606" y="513"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-65">\begin{array}{ll}(a+b)^{2}=a^{2}+2 a b+b^{2} & \left(2(a-b)^{2}=a^{2}-2 a b+b^{2}\right. \\ (a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} & (a)(a-b)^{3}=a^{3}-3 a^{2} b+3 a b^{2}-b^{3}\end{array}</script></div></div><hr /><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n72" cid="n72" mdtype="math_block">
			
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xlink:href="#E125-MJMATHI-62" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-32" x="606" y="583"></use></g><use xlink:href="#E125-MJMAIN-2B" x="13606" y="0"></use><use xlink:href="#E125-MJMAIN-22EF" x="14606" y="0"></use><use xlink:href="#E125-MJMAIN-2B" x="15945" y="0"></use></g><g transform="translate(0,-1680)"><g transform="translate(120,0)"><rect stroke="none" width="10169" height="60" x="0" y="220"></rect><g transform="translate(60,723)"><use xlink:href="#E125-MJMATHI-6E" x="0" y="0"></use><use xlink:href="#E125-MJMAIN-28" x="600" y="0"></use><use xlink:href="#E125-MJMATHI-6E" x="989" y="0"></use><use xlink:href="#E125-MJMAIN-2212" x="1811" y="0"></use><use xlink:href="#E125-MJMAIN-31" x="2811" y="0"></use><use xlink:href="#E125-MJMAIN-29" x="3311" y="0"></use><use xlink:href="#E125-MJMAIN-22EF" x="3867" y="0"></use><use xlink:href="#E125-MJMAIN-28" x="5205" y="0"></use><use xlink:href="#E125-MJMATHI-6E" x="5594" y="0"></use><use xlink:href="#E125-MJMAIN-2212" x="6417" y="0"></use><use xlink:href="#E125-MJMATHI-6B" x="7417" y="0"></use><use xlink:href="#E125-MJMAIN-2B" x="8160" y="0"></use><use xlink:href="#E125-MJMAIN-31" x="9160" y="0"></use><use xlink:href="#E125-MJMAIN-29" x="9660" y="0"></use></g><g transform="translate(4685,-690)"><use xlink:href="#E125-MJMATHI-6B" x="0" y="0"></use><use xlink:href="#E125-MJMAIN-21" x="521" y="0"></use></g></g><g transform="translate(10409,0)"><use xlink:href="#E125-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,412)"><use transform="scale(0.707)" xlink:href="#E125-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-2212" x="600" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMATHI-6B" x="1378" y="0"></use></g></g><g transform="translate(12381,0)"><use xlink:href="#E125-MJMATHI-62" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMATHI-6B" x="606" y="583"></use></g><use xlink:href="#E125-MJMAIN-2B" x="13501" y="0"></use><use xlink:href="#E125-MJMAIN-22EF" x="14501" y="0"></use><use xlink:href="#E125-MJMAIN-2B" x="15895" y="0"></use><use xlink:href="#E125-MJMATHI-6E" x="16895" y="0"></use><use xlink:href="#E125-MJMATHI-61" x="17495" y="0"></use><g transform="translate(18024,0)"><use xlink:href="#E125-MJMATHI-62" x="0" y="0"></use><g transform="translate(429,412)"><use transform="scale(0.707)" xlink:href="#E125-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-2212" x="600" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-31" x="1378" y="0"></use></g></g><use xlink:href="#E125-MJMAIN-2B" x="20103" y="0"></use><g transform="translate(21104,0)"><use xlink:href="#E125-MJMATHI-62" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMATHI-6E" x="606" y="583"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-68">\begin{aligned}(a+b)^{n}=\sum_{k=0}^{n} C_{n}^{k} a^{n-k} b^{k}=& a^{n}+n a^{n-1} b+\frac{n(n-1)}{2 !} a^{n-2} b^{2}+\cdots+\\ & \frac{n(n-1) \cdots(n-k+1)}{k !} a^{n-k} b^{k}+\cdots+n a b^{n-1}+b^{n} \end{aligned}</script></div></div><p><span>阶乘变换</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n82" cid="n82" mdtype="math_block">
			
		<div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1">
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xlink:href="#E126-MJMAIN-33" x="6712" y="0"></use><use xlink:href="#E126-MJMAIN-22C5" x="7434" y="0"></use><use xlink:href="#E126-MJMAIN-35" x="7934" y="0"></use><use xlink:href="#E126-MJMAIN-22C5" x="8657" y="0"></use><use xlink:href="#E126-MJMAIN-22EF" x="9157" y="0"></use><use xlink:href="#E126-MJMAIN-22C5" x="10551" y="0"></use><use xlink:href="#E126-MJMAIN-28" x="11051" y="0"></use><use xlink:href="#E126-MJMAIN-32" x="11440" y="0"></use><use xlink:href="#E126-MJMATHI-6E" x="11940" y="0"></use><use xlink:href="#E126-MJMAIN-2212" x="12763" y="0"></use><use xlink:href="#E126-MJMAIN-31" x="13763" y="0"></use><use xlink:href="#E126-MJMAIN-29" x="14263" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-64">\begin{array}{l}(2 n) ! !=2 \cdot 4 \cdot 6 \cdot \cdots \cdot(2 n)=2^{n} \cdot n ! \\ (2 n-1) ! !=1 \cdot 3 \cdot 5 \cdot \cdots \cdot(2 n-1)\end{array}</script>
					</div></div></blockquote><blockquote><p><span>不等式变换</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n99" cid="n99" mdtype="math_block">
			
		<div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1">
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id="MathJax-Element-84">\begin{array}{l}|a \pm b| \leqslant|a|+|b| ;<br>\\|| a|-| b|| \leqslant|a-b|;\\  \sqrt{a b} \leqslant \frac{a+b}{2} \leqslant \sqrt{\frac{a^{2}+b^{2}}{2}}(a, b>0) \\ \text { ( } \sqrt[3]{a b c} \leqslant \frac{a+b+c}{3} \leqslant \sqrt{\frac{a^{2}+b^{2}+c^{2}}{3}}(a, b, c>0) \\ 设  a>b>0 \text { ,则 }\left\{\begin{array}{l}\text { 当 } n>0 \text { 时 }, a^{n}>b^{n} \\ \text { 当 } n<0 \text { 时 }, a^{n}<b^{n}\end{array}\right. \\  0<a<x<b, 0<c<y<d \text { ,则 } \frac{c}{b}<\frac{y}{x}<\frac{d}{a} \text { . } \\  \sin x<x<\tan x\left(0<x<\frac{\pi}{2}\right) \\  \sin x<x(x>0) \\  \arctan x \leqslant x \leqslant \arcsin x(0 \leqslant x \leqslant 1) \\  \mathrm{e}^{x} \geqslant x+1(\forall x) \\  x-1 \geqslant \ln x(x>0)\\\frac{1}{1+x}<\ln \left(1+\frac{1}{x}\right)<\frac{1}{x}(x>0)\end{array}</script>
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